Continuity mistake: When Jamie is brought before Robb near the end, the blood smears on the sword being held to Jamie's neck keep changing between shots. (00:45:20)
Continuity mistake: When Jon is given the sword, his grip on the handle is constantly changing between shots. (00:11:55)
Answer: As Hand of the King, Tywin is the second most authoritative person in Westeros behind only the king. The Hand's duties include sitting on The Iron Throne and dispensing order, justice and judgment, similar to what Ned Stark did in the Season 1 episode "A Golden Crown" while then-king Robert Baratheon was out hunting. It's also worth pointing out that Tywin's enormous wealth makes him the de facto most powerful person in Westeros, which he demonstrates in the Season 3 episode "Mhysa" when he sends King Joffrey to bed without his supper and remarks to Tyrion "You're a fool if you believe he's the most powerful man in Westeros. Do you really think the crown gives you power?"